arXiv subjects

Michael W. Mahoney

Publications and source records attributed to Michael W. Mahoney.

At least 181 records · Page 10Linked to original sources

arXiv2021

Training Recommender Systems at Scale: Communication-Efficient Model and Data Parallelism

In this paper, we consider hybrid parallelism -- a paradigm that employs both Data Parallelism (DP) and Model Parallelism (MP) -- to scale distributed training of large recommendation models. We propose a compression framework called Dynamic Communication Thresholding (DCT) for communication-efficient hybrid training. DCT filters the entities to be communicated across the network through a simple hard-thresholding function, allowing only the most relevant information to pass through. For communication efficient DP, DCT compresses the parameter gradients sent to the parameter server during model synchronization. The threshold is updated only once every few thousand iterations to reduce the computational overhead of compression. For communication efficient MP, DCT incorporates a novel technique to compress the activations and gradients sent across the network during the forward and backward propagation, respectively. This is done by identifying and updating only the most relevant neurons of the neural network for each training sample in the data. We evaluate DCT on publicly available natural language processing and recommender models and datasets, as well as recommendation systems used in production at Facebook. DCT reduces communication by at least $100\times$ and $20\times$ during DP and MP, respectively. The algorithm has been deployed in production, and it improves end-to-end training time for a state-of-the-art industrial recommender model by 37\%, without any loss in performance.

cs.LG↗

arXiv2021

Block Basis Factorization for Scalable Kernel Matrix Evaluation

Kernel methods are widespread in machine learning; however, they are limited by the quadratic complexity of the construction, application, and storage of kernel matrices. Low-rank matrix approximation algorithms are widely used to address this problem and reduce the arithmetic and storage cost. However, we observed that for some datasets with wide intra-class variability, the optimal kernel parameter for smaller classes yields a matrix that is less well approximated by low-rank methods. In this paper, we propose an efficient structured low-rank approximation method -- the Block Basis Factorization (BBF) -- and its fast construction algorithm to approximate radial basis function (RBF) kernel matrices. Our approach has linear memory cost and floating-point operations for many machine learning kernels. BBF works for a wide range of kernel bandwidth parameters and extends the domain of applicability of low-rank approximation methods significantly. Our empirical results demonstrate the stability and superiority over the state-of-art kernel approximation algorithms.

stat.ML↗

arXiv2021

ADAHESSIAN: An Adaptive Second Order Optimizer for Machine Learning

We introduce ADAHESSIAN, a second order stochastic optimization algorithm which dynamically incorporates the curvature of the loss function via ADAptive estimates of the HESSIAN. Second order algorithms are among the most powerful optimization algorithms with superior convergence properties as compared to first order methods such as SGD and Adam. The main disadvantage of traditional second order methods is their heavier per iteration computation and poor accuracy as compared to first order methods. To address these, we incorporate several novel approaches in ADAHESSIAN, including: (i) a fast Hutchinson based method to approximate the curvature matrix with low computational overhead; (ii) a root-mean-square exponential moving average to smooth out variations of the Hessian diagonal across different iterations; and (iii) a block diagonal averaging to reduce the variance of Hessian diagonal elements. We show that ADAHESSIAN achieves new state-of-the-art results by a large margin as compared to other adaptive optimization methods, including variants of Adam. In particular, we perform extensive tests on CV, NLP, and recommendation system tasks and find that ADAHESSIAN: (i) achieves 1.80%/1.45% higher accuracy on ResNets20/32 on Cifar10, and 5.55% higher accuracy on ImageNet as compared to Adam; (ii) outperforms AdamW for transformers by 0.13/0.33 BLEU score on IWSLT14/WMT14 and 2.7/1.0 PPL on PTB/Wikitext-103; (iii) outperforms AdamW for SqueezeBert by 0.41 points on GLUE; and (iv) achieves 0.032% better score than Adagrad for DLRM on the Criteo Ad Kaggle dataset. Importantly, we show that the cost per iteration of ADAHESSIAN is comparable to first order methods, and that it exhibits robustness towards its hyperparameters.

cs.LG↗

arXiv2021

Lipschitz Recurrent Neural Networks

Viewing recurrent neural networks (RNNs) as continuous-time dynamical systems, we propose a recurrent unit that describes the hidden state's evolution with two parts: a well-understood linear component plus a Lipschitz nonlinearity. This particular functional form facilitates stability analysis of the long-term behavior of the recurrent unit using tools from nonlinear systems theory. In turn, this enables architectural design decisions before experimentation. Sufficient conditions for global stability of the recurrent unit are obtained, motivating a novel scheme for constructing hidden-to-hidden matrices. Our experiments demonstrate that the Lipschitz RNN can outperform existing recurrent units on a range of benchmark tasks, including computer vision, language modeling and speech prediction tasks. Finally, through Hessian-based analysis we demonstrate that our Lipschitz recurrent unit is more robust with respect to input and parameter perturbations as compared to other continuous-time RNNs.

cs.LG↗

arXiv2021

Adversarially-Trained Deep Nets Transfer Better: Illustration on Image Classification

Transfer learning has emerged as a powerful methodology for adapting pre-trained deep neural networks on image recognition tasks to new domains. This process consists of taking a neural network pre-trained on a large feature-rich source dataset, freezing the early layers that encode essential generic image properties, and then fine-tuning the last few layers in order to capture specific information related to the target situation. This approach is particularly useful when only limited or weakly labeled data are available for the new task. In this work, we demonstrate that adversarially-trained models transfer better than non-adversarially-trained models, especially if only limited data are available for the new domain task. Further, we observe that adversarial training biases the learnt representations to retaining shapes, as opposed to textures, which impacts the transferability of the source models. Finally, through the lens of influence functions, we discover that transferred adversarially-trained models contain more human-identifiable semantic information, which explains -- at least partly -- why adversarially-trained models transfer better.

cs.LG↗

arXiv2021

Hessian Eigenspectra of More Realistic Nonlinear Models

Given an optimization problem, the Hessian matrix and its eigenspectrum can be used in many ways, ranging from designing more efficient second-order algorithms to performing model analysis and regression diagnostics. When nonlinear models and non-convex problems are considered, strong simplifying assumptions are often made to make Hessian spectral analysis more tractable. This leads to the question of how relevant the conclusions of such analyses are for more realistic nonlinear models. In this paper, we exploit deterministic equivalent techniques from random matrix theory to make a \emph{precise} characterization of the Hessian eigenspectra for a broad family of nonlinear models, including models that generalize the classical generalized linear models, without relying on strong simplifying assumptions used previously. We show that, depending on the data properties, the nonlinear response model, and the loss function, the Hessian can have \emph{qualitatively} different spectral behaviors: of bounded or unbounded support, with single- or multi-bulk, and with isolated eigenvalues on the left- or right-hand side of the bulk. By focusing on such a simple but nontrivial nonlinear model, our analysis takes a step forward to unveil the theoretical origin of many visually striking features observed in more complex machine learning models.

stat.ML↗

arXiv2021

Improving Semi-supervised Federated Learning by Reducing the Gradient Diversity of Models

Federated learning (FL) is a promising way to use the computing power of mobile devices while maintaining the privacy of users. Current work in FL, however, makes the unrealistic assumption that the users have ground-truth labels on their devices, while also assuming that the server has neither data nor labels. In this work, we consider the more realistic scenario where the users have only unlabeled data, while the server has some labeled data, and where the amount of labeled data is smaller than the amount of unlabeled data. We call this learning problem semi-supervised federated learning (SSFL). For SSFL, we demonstrate that a critical issue that affects the test accuracy is the large gradient diversity of the models from different users. Based on this, we investigate several design choices. First, we find that the so-called consistency regularization loss (CRL), which is widely used in semi-supervised learning, performs reasonably well but has large gradient diversity. Second, we find that Batch Normalization (BN) increases gradient diversity. Replacing BN with the recently-proposed Group Normalization (GN) can reduce gradient diversity and improve test accuracy. Third, we show that CRL combined with GN still has a large gradient diversity when the number of users is large. Based on these results, we propose a novel grouping-based model averaging method to replace the FedAvg averaging method. Overall, our grouping-based averaging, combined with GN and CRL, achieves better test accuracy than not just a contemporary paper on SSFL in the same settings (>10\%), but also four supervised FL algorithms.

cs.LG↗

arXiv2021

Good Classifiers are Abundant in the Interpolating Regime

Within the machine learning community, the widely-used uniform convergence framework has been used to answer the question of how complex, over-parameterized models can generalize well to new data. This approach bounds the test error of the worst-case model one could have fit to the data, but it has fundamental limitations. Inspired by the statistical mechanics approach to learning, we formally define and develop a methodology to compute precisely the full distribution of test errors among interpolating classifiers from several model classes. We apply our method to compute this distribution for several real and synthetic datasets, with both linear and random feature classification models. We find that test errors tend to concentrate around a small typical value $\varepsilon^*$, which deviates substantially from the test error of the worst-case interpolating model on the same datasets, indicating that "bad" classifiers are extremely rare. We provide theoretical results in a simple setting in which we characterize the full asymptotic distribution of test errors, and we show that these indeed concentrate around a value $\varepsilon^*$, which we also identify exactly. We then formalize a more general conjecture supported by our empirical findings. Our results show that the usual style of analysis in statistical learning theory may not be fine-grained enough to capture the good generalization performance observed in practice, and that approaches based on the statistical mechanics of learning may offer a promising alternative.

stat.ML↗

arXiv2021

A Differential Geometry Perspective on Orthogonal Recurrent Models

Recently, orthogonal recurrent neural networks (RNNs) have emerged as state-of-the-art models for learning long-term dependencies. This class of models mitigates the exploding and vanishing gradients problem by design. In this work, we employ tools and insights from differential geometry to offer a novel perspective on orthogonal RNNs. We show that orthogonal RNNs may be viewed as optimizing in the space of divergence-free vector fields. Specifically, based on a well-known result in differential geometry that relates vector fields and linear operators, we prove that every divergence-free vector field is related to a skew-symmetric matrix. Motivated by this observation, we study a new recurrent model, which spans the entire space of vector fields. Our method parameterizes vector fields via the directional derivatives of scalar functions. This requires the construction of latent inner product, gradient, and divergence operators. In comparison to state-of-the-art orthogonal RNNs, our approach achieves comparable or better results on a variety of benchmark tasks.

cs.LG↗

arXiv2021

Noise-Response Analysis of Deep Neural Networks Quantifies Robustness and Fingerprints Structural Malware

The ubiquity of deep neural networks (DNNs), cloud-based training, and transfer learning is giving rise to a new cybersecurity frontier in which unsecure DNNs have `structural malware' (i.e., compromised weights and activation pathways). In particular, DNNs can be designed to have backdoors that allow an adversary to easily and reliably fool an image classifier by adding a pattern of pixels called a trigger. It is generally difficult to detect backdoors, and existing detection methods are computationally expensive and require extensive resources (e.g., access to the training data). Here, we propose a rapid feature-generation technique that quantifies the robustness of a DNN, `fingerprints' its nonlinearity, and allows us to detect backdoors (if present). Our approach involves studying how a DNN responds to noise-infused images with varying noise intensity, which we summarize with titration curves. We find that DNNs with backdoors are more sensitive to input noise and respond in a characteristic way that reveals the backdoor and where it leads (its `target'). Our empirical results demonstrate that we can accurately detect backdoors with high confidence orders-of-magnitude faster than existing approaches (seconds versus hours).

cs.LG↗

arXiv2021

Boundary thickness and robustness in learning models

Robustness of machine learning models to various adversarial and non-adversarial corruptions continues to be of interest. In this paper, we introduce the notion of the boundary thickness of a classifier, and we describe its connection with and usefulness for model robustness. Thick decision boundaries lead to improved performance, while thin decision boundaries lead to overfitting (e.g., measured by the robust generalization gap between training and testing) and lower robustness. We show that a thicker boundary helps improve robustness against adversarial examples (e.g., improving the robust test accuracy of adversarial training) as well as so-called out-of-distribution (OOD) transforms, and we show that many commonly-used regularization and data augmentation procedures can increase boundary thickness. On the theoretical side, we establish that maximizing boundary thickness during training is akin to the so-called mixup training. Using these observations, we show that noise-augmentation on mixup training further increases boundary thickness, thereby combating vulnerability to various forms of adversarial attacks and OOD transforms. We can also show that the performance improvement in several lines of recent work happens in conjunction with a thicker boundary.

cs.LG↗

arXiv2020

Shallow Neural Networks for Fluid Flow Reconstruction with Limited Sensors

In many applications, it is important to reconstruct a fluid flow field, or some other high-dimensional state, from limited measurements and limited data. In this work, we propose a shallow neural network-based learning methodology for such fluid flow reconstruction. Our approach learns an end-to-end mapping between the sensor measurements and the high-dimensional fluid flow field, without any heavy preprocessing on the raw data. No prior knowledge is assumed to be available, and the estimation method is purely data-driven. We demonstrate the performance on three examples in fluid mechanics and oceanography, showing that this modern data-driven approach outperforms traditional modal approximation techniques which are commonly used for flow reconstruction. Not only does the proposed method show superior performance characteristics, it can also produce a comparable level of performance with traditional methods in the area, using significantly fewer sensors. Thus, the mathematical architecture is ideal for emerging global monitoring technologies where measurement data are often limited.

physics.comp-ph↗

arXiv2020

A Random Matrix Analysis of Random Fourier Features: Beyond the Gaussian Kernel, a Precise Phase Transition, and the Corresponding Double Descent

This article characterizes the exact asymptotics of random Fourier feature (RFF) regression, in the realistic setting where the number of data samples $n$, their dimension $p$, and the dimension of feature space $N$ are all large and comparable. In this regime, the random RFF Gram matrix no longer converges to the well-known limiting Gaussian kernel matrix (as it does when $N \to \infty$ alone), but it still has a tractable behavior that is captured by our analysis. This analysis also provides accurate estimates of training and test regression errors for large $n,p,N$. Based on these estimates, a precise characterization of two qualitatively different phases of learning, including the phase transition between them, is provided; and the corresponding double descent test error curve is derived from this phase transition behavior. These results do not depend on strong assumptions on the data distribution, and they perfectly match empirical results on real-world data sets.

stat.ML↗

arXiv2020

Improved guarantees and a multiple-descent curve for Column Subset Selection and the Nyström method

The Column Subset Selection Problem (CSSP) and the Nyström method are among the leading tools for constructing small low-rank approximations of large datasets in machine learning and scientific computing. A fundamental question in this area is: how well can a data subset of size k compete with the best rank k approximation? We develop techniques which exploit spectral properties of the data matrix to obtain improved approximation guarantees which go beyond the standard worst-case analysis. Our approach leads to significantly better bounds for datasets with known rates of singular value decay, e.g., polynomial or exponential decay. Our analysis also reveals an intriguing phenomenon: the approximation factor as a function of k may exhibit multiple peaks and valleys, which we call a multiple-descent curve. A lower bound we establish shows that this behavior is not an artifact of our analysis, but rather it is an inherent property of the CSSP and Nyström tasks. Finally, using the example of a radial basis function (RBF) kernel, we show that both our improved bounds and the multiple-descent curve can be observed on real datasets simply by varying the RBF parameter.

cs.LG↗

arXiv2020

A Statistical Framework for Low-bitwidth Training of Deep Neural Networks

Fully quantized training (FQT), which uses low-bitwidth hardware by quantizing the activations, weights, and gradients of a neural network model, is a promising approach to accelerate the training of deep neural networks. One major challenge with FQT is the lack of theoretical understanding, in particular of how gradient quantization impacts convergence properties. In this paper, we address this problem by presenting a statistical framework for analyzing FQT algorithms. We view the quantized gradient of FQT as a stochastic estimator of its full precision counterpart, a procedure known as quantization-aware training (QAT). We show that the FQT gradient is an unbiased estimator of the QAT gradient, and we discuss the impact of gradient quantization on its variance. Inspired by these theoretical results, we develop two novel gradient quantizers, and we show that these have smaller variance than the existing per-tensor quantizer. For training ResNet-50 on ImageNet, our 5-bit block Householder quantizer achieves only 0.5% validation accuracy loss relative to QAT, comparable to the existing INT8 baseline.

cs.LG↗

arXiv2020

MAF: Multimodal Alignment Framework for Weakly-Supervised Phrase Grounding

Phrase localization is a task that studies the mapping from textual phrases to regions of an image. Given difficulties in annotating phrase-to-object datasets at scale, we develop a Multimodal Alignment Framework (MAF) to leverage more widely-available caption-image datasets, which can then be used as a form of weak supervision. We first present algorithms to model phrase-object relevance by leveraging fine-grained visual representations and visually-aware language representations. By adopting a contrastive objective, our method uses information in caption-image pairs to boost the performance in weakly-supervised scenarios. Experiments conducted on the widely-adopted Flickr30k dataset show a significant improvement over existing weakly-supervised methods. With the help of the visually-aware language representations, we can also improve the previous best unsupervised result by 5.56%. We conduct ablation studies to show that both our novel model and our weakly-supervised strategies significantly contribute to our strong results.

cs.CL↗

arXiv2020

Sparse Quantized Spectral Clustering

Given a large data matrix, sparsifying, quantizing, and/or performing other entry-wise nonlinear operations can have numerous benefits, ranging from speeding up iterative algorithms for core numerical linear algebra problems to providing nonlinear filters to design state-of-the-art neural network models. Here, we exploit tools from random matrix theory to make precise statements about how the eigenspectrum of a matrix changes under such nonlinear transformations. In particular, we show that very little change occurs in the informative eigenstructure even under drastic sparsification/quantization, and consequently that very little downstream performance loss occurs with very aggressively sparsified or quantized spectral clustering. We illustrate how these results depend on the nonlinearity, we characterize a phase transition beyond which spectral clustering becomes possible, and we show when such nonlinear transformations can introduce spurious non-informative eigenvectors.

stat.ML↗

arXiv2020

OverSketched Newton: Fast Convex Optimization for Serverless Systems

Motivated by recent developments in serverless systems for large-scale computation as well as improvements in scalable randomized matrix algorithms, we develop OverSketched Newton, a randomized Hessian-based optimization algorithm to solve large-scale convex optimization problems in serverless systems. OverSketched Newton leverages matrix sketching ideas from Randomized Numerical Linear Algebra to compute the Hessian approximately. These sketching methods lead to inbuilt resiliency against stragglers that are a characteristic of serverless architectures. Depending on whether the problem is strongly convex or not, we propose different iteration updates using the approximate Hessian. For both cases, we establish convergence guarantees for OverSketched Newton and empirically validate our results by solving large-scale supervised learning problems on real-world datasets. Experiments demonstrate a reduction of ~50% in total running time on AWS Lambda, compared to state-of-the-art distributed optimization schemes.

cs.DC↗